1. CSE 321 Discrete Structures
Winter 2008
Lecture 20
Probability: Bernoulli, Random Variables, Bayes’ Theorem
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2. Announcements Readings Probability Theory
6.3 (New material!) Bayes’ Theorem
6.4 (5.3) Expectation
Advanced Counting Techniques – Ch 7.
Not covered
Relations
Chapter 8 (Chapter 7)

3. Highlights from Lecture 19
Conditional Probability
Independence
Let E and F be events with p(F) > 0. The conditional probability of E given F, defined by p(E | F), is defined as:
The events E and F are independent if and only if
p(E F) = p(E)p(F)

4. Bernoulli Trials and Binomial Distribution
Success probability p, failure probability q
The probability of exactly k successes in n
independent Bernoulli trials is

5. Random Variables A random variable is a function from
a sample space to the real numbers

6. Bayes’ Theorem Suppose that E and F are events from a sample
space S such that p(E) > 0 and p(F) > 0. Then

7. False Positives, False Negatives
Let D be the event that a person has the disease
Let Y be the event that a person tests positive
for the disease

8. Testing for disease Disease is very rare: p(D) = 1/100,000
Testing is accurate:
False negative: 1%
False positive: 0.5%
Suppose you get a positive result, what
do you conclude?
P(YC|D)
P(Y|DC)

9. P(D|Y)
Answer is about 0.002

10. Spam Filtering
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11. Bayesian Spam filters Classification domain Criteria for spam
Cost of false negative
Cost of false positive
Criteria for spam
v1agra, ONE HUNDRED MILLION USD
Basic question: given an message, based on spam criteria, what is the probability it is spam

12. Email message with phrase “Account Review”
250 of messages known to be spam
5 of messages known not to be spam
Assuming 50% of messages are spam, what is the probability that a message with “Account Review” is spam

13. Proving Bayes’ Theorem

14. Expectation The expected value of random variable X(s) on
sample space S is:

15. Expectation examples Number of heads when flipping a coin 3 times
Sum of two dice
Successes in n Bernoulli trials with success probability p